Meromorphic Solutions to Higher Order Nonlinear Delay Differential Equations

Let w(z) be non-rational meromorphic solutions with hyper-order less than 1 to a family of higher order nonlinear delay differential equations

$$\matrix{{w\left( {z + 1} \right)w\left( {z - 1} \right)\, + \,a\left( z \right){{{w^{\left( k \right)}}\left( z \right)} \over {w\left( z \right)}} = R\left( {z,\,w\left( z \right)} \right),} & {k \in \mathbb{N}{^ + },} \cr}$$

where a(z) is rational, \(R\left( {z,\,w\left( z \right)} \right) = {{P\left( {z,\,w,\,\left( z \right)} \right)} \over {Q\left( {z,\,w,\,\left( z \right)} \right)}}\) is an irreducible rational function in w with rational coefficients in z. This paper mainly show the relationships of the degree of P(z,w(z)) and Q(z,w(z)) when the above equations exist such solutions w(z). There are also some examples to show that our results are sharp.